Saddlepoint approximations for the normalizing constant of Fisher-Bingham distributions on products of spheres and Stiefel manifolds

In an earlier paper Kume & Wood (2005) showed how the normalizing constant of the Fisher–\udBingham distribution on a sphere can be approximated with high accuracy using a univariate saddlepoint\uddensity approximation. In this sequel, we extend the approach to a more general setting\udand derive saddlepoint approximations for the normalizing constants of multicomponent Fisher–\udBingham distributions on Cartesian products of spheres, and Fisher–Bingham distributions on\udStiefel manifolds. In each case, the approximation for the normalizing constant is essentially\uda multivariate saddlepoint density approximation for the joint distribution of a set of quadratic\udforms in normal variables. Both first-order and second-order saddlepoint approximations are considered.\udComputational algorithms, numerical results and theoretical properties of the approximations\udare presented. In the challenging high-dimensional settings considered in this paper the\udsaddlepoint approximations perform very well in all examples considered.\udSome key words: Directional data; Fisher matrix distribution; Kent distribution; Orientation statistics